A curvature-based approach for left ventricular shape analysis from cardiac magnetic resonance imaging
Abstract
It is believed that left ventricular (LV) regional shape is indicative of LV regional function, and cardiac pathologies are often associated with regional alterations in ventricular shape. In this article, we present a set of procedures for evaluating regional LV surface shape from anatomically accurate models reconstructed from cardiac magnetic resonance (MR) images. LV surface curvatures are computed using local surface fitting method, which enables us to assess regional LV shape and its variation. Comparisons are made between normal and diseased hearts. It is illustrated that LV surface curvatures at different regions of the normal heart are higher than those of the diseased heart. Also, the normal heart experiences a larger change in regional curvedness during contraction than the diseased heart. It is believed that with a wide range of dataset being evaluated, this approach will provide a new and efficient way of quantifying LV regional function.
Keywords
Left ventricular shape Surface curvature Local surface fitting MRI Regional function1 Introduction
The shape of the left ventricle (LV) has intrigued physiologists attempting to gain a better understanding of its mode of operation, as well as clinicians trying to obtain diagnostic information on its performance [14, 25]. In general, the LV responds to volume overload by dilating to a more spherical shape and to pressure overload with hypertrophy in patients with ischemic cardiomyopathy. Furthermore, surgical procedures (e.g., the Dor procedure) can enable restoration of LV shape [15]. Accordingly, the shape of the LV is an important diagnostic and therapeutic index for evaluating a variety of cardiac disease states. Furthermore, knowledge of the LV remodeling process is important to (1) better understand and track the LV response to pressure and volume overload, (2) elucidate the mechanism of LV dysfunction in cardiac conditions that affect the chamber, and (3) better correlate patient symptoms to the underlying pathophysiology. In view of the association of LV shape with cardiac disease states and LV dysfunction, we aimed to develop a method for three-dimensional shape analysis of the LV by combining MRI and computational methods.
In the previous studies, the shape analysis methods were mainly on two-dimensional (2D) tomographic section (eg, 2D echocardiography and magnetic resonance imaging) of the heart using simple indices [12, 38] or three-dimensional (3D) echocardiography using sophistical curvature analysis [17]. There are, however, various shortcomings in using echocardiography [31], ventriculography [22], angiography [33] and indicator-dilution methods [13] compared with magnetic resonance imaging (MRI).
In recent years, computerized image segmentation has played an increasingly important role in the field of medical imaging, and has been applied in various applications [2, 4, 11]. Some popular methods used for medical image segmentation include thresholding [30], edge-based methods [26], deformable models [24] and level-sets methods [6]. Although these methods provide an automatic approach to image segmentation, they often produce problems such as “leaking” through weak and diffuse edges, and over-segmentation due to the excessive number of local minima when proper “denoising” is not provided [6, 26].
In fact, because of the relatively large spacing between the MR image slices (typically 5–10 mm for cardiac MRI in clinical practices) and the inhomogeneous contrast of the myocardium, these segmentation methods may not be suitable for the delineation of the LV contours [6, 26]. Also, existing automatic segmentation methods for CMR images are often based on algorithms that are sensitive to image quality and frequently depend on the specific imaging protocol [1, 10, 19, 29]. Therefore, in this work, the segmentation of LV endocardial contours has been performed manually by a trained expert using CMRtools, which is a software package for the display and quantitative analysis of cardiac MR images [32].
Existing methods for estimating higher order properties, such as curvatures, of discrete surface information (e.g., triangulated mesh) can be classified into three general categories. The first group of methods computes the curvature directly from the triangulation by using the concept of discrete differential geometry [5, 18]. Although these methods involve low computational cost, their accuracies are largely affected by the quality of the triangulation, and they often become unstable near degenerate configurations. The second group of methods uses an analytical approach to extract the differential properties and curvature information by means of local surface fitting [36, 37, 39]. In general, a quadratic polynomial function is used to approximate a smooth surface to a local set of points in a region. These methods are generally robust and produce accurate results as shown by Garimella and Swartz [21], and Cazals and Pouget [7]. The third group of methods extracts curvature information by computing the average of the curvature tensor over a small area of the triangulated surface [3, 35]. These tensor-based methods often produce significant errors in estimating curvature, even for densely triangulated models and simple geometries such as a sphere or ellipsoid.
In this paper, we have chosen to characterize the LV shape by deriving the LV surface curvatures via an analytical approach using a surface patch fitting method, due to its robustness and accuracy. We have applied this method to five normal subjects and five heart disease patients.
2 Methods
2.1 MRI scans
This study involved five normal subjects and five patients with heart diseases. All subjects underwent diagnostic MRI scan. None of the normal subjects had (1) significant valvular or congenital cardiac disease, (2) history of myocardial infarction, (3) coronary artery lesions, or (4) abnormal left ventricular pressure, end-diastolic volume or ejection fraction.
2.2 Data processing and geometry reconstruction
Due to the relatively large spacing between the cardiac MR image slices as well as the contrast inhomogeneity of the myocardial tissue, it is not efficient to segment the LV contours using automatic methods. Instead, the segmentation of LV endocardial contours is performed manually by a trained expert using CMRtools.
2.3 Computation of three-dimensional shape descriptor
In this paper, the intrinsic LV surface properties are computed via an analytical approach, using a surface patch fitting method. Such surface fitting methods have been used by researchers to investigate the surface curvatures of anatomical structures of the human body, and also to define anatomical landmarks for various applications [16, 28].
2.3.1 Local surface patch fitting
The index C is a positive value, and describes the magnitude of the curvedness at a point, and thus is a measure of how gently or highly curved a surface is.
2.3.2 Computation of LV surface curvatures
- 1.
Interactively segment the LV contours from the cardiac MR images.
- 2.
Reconstruct the LV surface by triangulation of the LV contour points derived from the segmentation process.
- 3.
For each vertex, select the n-ring neighboring vertices.
- 4.
At each selected vertex, perform quadratic surface patch fitting and compute the surface curvature.
The choice of the n-ring neighboring points used for the patch fitting is arbitrary. However as shown in the previous section, there are six unknowns c_{1}–c_{6} in the quadratic function. Therefore, at least five neighboring points around the selected vertex are required for the local patch fitting, and a 1-ring neighborhood may produce errors in regions with insufficient points. On the other hand, selecting a region with too many points (eg, a 5-ring neighborhood) will cause an “over-smoothing” of the surface patch being considered. Therefore in this work, local surface fitting and curvature computation at each vertex is performed based on a set of 2-ring neighboring points.
3 Results
Characteristics of normal and heart failure subjects
Normal (n = 5) | Heart disease (n = 5) | p value | |
---|---|---|---|
Age (years) | 34 ± 24 | 51 ± 10 | 0.17 |
Weight (kg) | 71 ± 20 | 68 ± 18 | 0.81 |
Height (cm) | 169 ± 10 | 160 ± 7 | 0.15 |
Diastolic blood pressure (mmHg) | 74 ± 9 | 72 ± 14 | 0.79 |
Systolic blood pressure (mmHg) | 133 ± 11 | 108 ± 8 | 0.0026 |
HR (beats/min) | 70 ± 8 | 75 ± 19 | 0.59 |
CI (ml/m^{2}) | 3.3 ± 0.2 | 2.1 ± 0.2 | <0.001 |
EDVI (ml/m^{2}) | 68 ± 7 | 146 ± 20 | <0.001 |
ESVI (ml/m^{2}) | 24 ± 5 | 119 ± 15 | <0.001 |
EF (%) | 65 ± 5 | 18 ± 4 | <0.001 |
LV regional curvedness, C_{Rgn} (mm^{−1}) at end-diastolic (ED) and end-systolic (ES) phases, and the associated percent regional curvedness phase-change (ΔC_{Rgn}) in the 16-segment heart model for normal and diseased heart subjects
Segment | Normal (n = 5) | Diseased (n = 5) | ||||
---|---|---|---|---|---|---|
C_{Rgn,ED} (mm^{−1}) | C_{Rgn,ES} (mm^{−1}) | ∆C_{Rgn} (%) | C_{Rgn,ED} (mm^{−1}) | C_{Rgn,ES} (mm^{−1}) | ∆C_{Rgn} (%) | |
Basal anterior | 0.045 ± 0.0052 | 0.066 ± 0.0052 | 32 ± 9 | 0.044 ± 0.025 | 0.048 ± 0.024 | 9 ± 9* |
Basal anteroseptal | 0.044 ± 0.0070 | 0.065 ± 0.0048 | 32 ± 9 | 0.040 ± 0.013 | 0.046 ± 0.016 | 12 ± 8* |
Basal inferoseptal | 0.037 ± 0.0061 | 0.054 ± 0.0066 | 33 ± 10 | 0.041 ± 0.018 | 0.041 ± 0.018 | 2 ± 18* |
Basal inferior | 0.047 ± 0.0074 | 0.063 ± 0.0056 | 26 ± 8 | 0.043 ± 0.018 | 0.043 ± 0.022 | −3 ± 26* |
Basal inferolateral | 0.036 ± 0.0066 | 0.056 ± 0.0081 | 35 ± 3 | 0.038 ± 0.019 | 0.041 ± 0.021 | 7 ± 21* |
Basal anterolateral | 0.036 ± 0.0048 | 0.056 ± 0.0063 | 34 ± 15 | 0.044 ± 0.026 | 0.041 ± 0.025 | −8 ± 17* |
Mid anterior | 0.051 ± 0.016 | 0.075 ± 0.014 | 37 ± 13 | 0.041 ± 0.017 | 0.045 ± 0.013* | 15 ± 12* |
Mid anteroseptal | 0.054 ± 0.015 | 0.091 ± 0.024 | 40 ± 14 | 0.037 ± 0.007 | 0.042 ± 0.004* | 15 ± 17* |
Mid inferoseptal | 0.045 ± 0.0099 | 0.081 ± 0.0140 | 44 ± 14 | 0.046 ± 0.018 | 0.049 ± 0.012* | 9 ± 25* |
Mid inferior | 0.057 ± 0.016 | 0.084 ± 0.015 | 31 ± 17 | 0.046 ± 0.017 | 0.053 ± 0.013* | 12 ± 30* |
Mid inferolateral | 0.047 ± 0.0093 | 0.073 ± 0.020 | 32 ± 22 | 0.039 ± 0.012 | 0.042 ± 0.007* | 8 ± 15* |
Mid anterolateral | 0.044 ± 0.0010 | 0.065 ± 0.020 | 34 ± 16 | 0.044 ± 0.017 | 0.041 ± 0.021 | −15 ± 21* |
Apical anterior | 0.087 ± 0.0145 | 0.215 ± 0.093 | 56 ± 11 | 0.049 ± 0.015* | 0.053 ± 0.010* | 8 ± 15* |
Apical septal | 0.110 ± 0.042 | 0.209 ± 0.082 | 48 ± 13 | 0.044 ± 0.008* | 0.049 ± 0.008* | 10 ± 9* |
Apical inferior | 0.124 ± 0.044 | 0.243 ± 0.085 | 48 ± 14 | 0.058 ± 0.016* | 0.062 ± 0.011* | 13 ± 6* |
Apical lateral | 0.108 ± 0.039 | 0.272 ± 0.127 | 56 ± 20 | 0.056 ± 0.010* | 0.059 ± 0.010* | 10 ± 16* |
It is found that ∆C_{Rgn} for different regions of the normal heart are much higher than those found for corresponding regions of the diseased heart. This relatively large difference in ΔC_{Rgn} is due to the large surface deformation (due to good LV contraction during systole) of the normal heart, as compared to the minimally deformed diseased heart (due to impaired LV function) during contraction. For example, the mean values of ΔC_{Rgn} at the apical region of the diseased heart are found to be 8, 10, 13 and 10% at the apical anterior, septal, inferior and lateral segments, respectively, which are lower than those observed for the normal heart. Note the negative ΔC_{Rgn} (i.e., −15%) found at the mid anterolateral segment of the diseased heart, this indicates that instead of becoming more curved during the systolic phase, this region of the LV becomes flatter. These lower and negative ΔC_{Rgn} values found in the diseased heart may be attributed to the impaired LV systolic function and regional contractile function of the infarcted myocardial segments.
4 Discussion
In this paper, the differential properties and curvature information of the LV surface are extracted via an analytical approach by means of local surface fitting [36, 39]. This approach is robust and produces accurate results, as shown by Garimella and Swartz [21], and Cazals and Pouget [7]. Here, we examine the LV shape derived from anatomically accurate reconstructed 3D models of the LV, and are able to assess LV regional shape in terms of curvatures. This is in contrast to previous studies, which limited the analysis to simplified shape indices or parameters and 2D models based on LV short- and long-axes [23, 38, 40].
The Gaussian (K) and mean curvatures (H) are considered the most widely used indicators for surface shape classification. Given a point on a surface, K is used to categorize the point into elliptic, hyperbolic, parabolic or planar based on the sign of K, while H can be used to differentiate concave (i.e., −ve H) regions from convex (i.e., +ve H) regions. However, Koenderink and van Doorn [27] have shown that K and H are not very indicative of local shape, and introduced more significant measures of local shape known as the shape index (S) and curvedness (C). In their formulation, S is scale invariant and provides a smooth categorization of shapes, such as concave shapes (−1 < S < − 1/2), hyperboloid shapes (−1/2 < S < 1/2) and convex shapes (1/2 < S < 1). The curvedness C describes the magnitude of the curvature at a surface point. It is a measure of how highly or gently curved a point is.
Since we are interested in the magnitude or extent of curvature changes (i.e., surface deformation as the LV contracts) and not the categorization of shapes or directions of curvatures, we chose to use C as the curvature metric for analyzing LV regional function. This metric has an advantage over H, in the situation when the mean curvature value vanishes at surface points where κ_{1} = −κ_{2} and its magnitude is not intuitive (i.e., H is small although surface is highly curved with κ_{1} ≈ −κ_{2}, since H is an average of the principal curvatures). Also, at surface regions where the H values between two time frames have the same magnitude but differ in sign convention, ∆H gives a value of zero even though a significant deformation is involved.
In this study, we are able to elucidate and assess the LV surface shape and its deformation in terms of curvatures. The results show that the normal LV has significantly higher surface curvedness (i.e., high C value) as compared to the diseased heart (i.e., low C value) at the end-systolic phase. This is due to the relatively greater curved surface of the normal LV chamber in contrast to the dilated LV chamber of the diseased heart. We have also demonstrated that the normal LV experiences a significant increase in curvedness (i.e., high ΔC_{Rgn}) when it contracts during systole, while the diseased LV undergoes minimal change in curvedness (i.e., low ΔC_{Rgn}) during systole. This considerable large difference in ΔC_{Rgn} between the normal and diseased hearts illustrates that the normal LV contracts and deforms more than the diseased LV. This finding also provides testimony to the sensitivity and accuracy of our 3D LV shape reconstruction methodology and our application of the surface curvedness indicators.
The CSAtools can be used to read in segmented contour points, to reconstruct the LV surface, compute and visualize LV curvature distribution, and analyze the regional LV curvatures. This set of procedures (i.e., from input contour points to output regional curvatures) takes only a few minutes to perform, and thus provides an efficient and convenient tool for clinical diagnosis. This is in contrast to methods using finite element simulation of LV wall deformation and myocardial stress/strain analysis, which usually take several hours or even days for the simulation to converge. In addition, the percent curvedness phase-change (∆C) plots in Table 2 enable us to assess LV regional function easily, which may be a valuable tool for detecting unusual LV wall activities as well as ischemic and infarcted myocardial segments.
There are many clinical implications based on the assessment of LV shape indices. By studying the surface curvedness of the LV endocardial surface during the cardiac cycle, one can get a better perception of the regional wall deformation of the LV. Also, the knowledge of normal values of regional curvedness and percent curvedness change in the LV may allow the identification and follow-up of local functional abnormities. We envisage that with a wide range of LV datasets being tested and evaluated, we can easily correlate the changes in LV surface curvedness with the intricate LV regional function. This will eventually be useful for the assessment of regional cardiac function, especially for the detection of infarct regions and unusual wall activities.
4.1 Limitations
The accuracy of this method depends on factors such as image resolution and the surface reconstruction process. The spacing between short-axis image slices for cardiac MRI in clinical practices is typically around 5–10 mm. Therefore, a considerable amount of interpolation between image slices has been used, and this may affect the accuracy of the curvature evaluation. The accuracy of the surface curvatures also depends on the mesh quality and densities of the LV models, since the differential properties of each surface point are computed using the neighboring vertices. Due to the resolution of the MR images used and a relatively more intricate shape of the apical region, segmentation of LV contours is often more difficult in the apical region than in the mid and basal regions. This could affect the evaluation of the regional curvedness at the apical region, especially at ES. In this study, five healthy LVs and five diseased LVs have been evaluated. A larger range of dataset needs to be assessed so that one can get a more accurate correlation between LV curvature changes and LV regional function (i.e., ventricular wall motion). Currently, more work is being done in this direction.
5 Conclusions
We have developed a framework for the rapid and accurate assessment of LV shape and its associated changes during the cardiac cycle. The set of procedures pertain to (1) the segmentation and reconstruction of anatomically accurate LV models from cardiac MR images, and (2) the extraction of differential properties of the LV surface via analytic surface fitting technique. This enables us to analyze LV regional shape and deformation efficiently and accurately, in terms of surface curvedness. The regional curvedness of different LV segments and their associated percent curvedness phase-changes are evaluated for normal and diseased hearts, using the above-mentioned methodology. It is shown that the regional surface curvedness and their changes (from ED to ES) for the normal heart differ greatly from those of the diseased heart. Thus, this new approach may be an efficient way of quantifying LV regional function.
Notes
Acknowledgments
This work was supported in part by a research grant from the National Heart Centre, Singapore.
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